A look at the list of UFO sightings from Wikipedia shows the "UFO craze" beginning in 1947. Of course there were previous sightings of "unidentified flying object"; a duck is an UFO if whoever sees it does not recognize it as a duck. But beginning in 1947, the number of reported sightings just jumped. The notion of sentient extra-terrestrial life forms, possibly hostile to mankind, was well established and mainstream at the end of the 19th century; apart from H. G. Wells' book, as @H.R.Rambler indicates, one may cite Lowell's Martian canals. The 1938 panic triggered by Orson Welles shows that by 1938 the public at large was ready to believe in aliens. Yet there can be a difference between a fertile situation, and an actual "craze".
Personally I would define the UFO craze to begin in 1947, and to end around 1990; basically when movies began to have sufficiently convincing special effects that all reported photographs of flying saucers could instinctively be thought of as "potentially phony" by spectators. The X-Files television series demonstrates, in my opinion, that the craze was ended: instead of people claiming "I want to believe", they made a fictional show about a hero who "wants to believe", thereby precipitating the whole thing into the category of "funny irrelevance".
Among the same lines, the "fairy craze" ended much before the "UFO craze" began. In the 1930s, J.R.R. Tolkien could use the fairy worlds and inhabitants for fiction, even children's books (as The Hobbit was supposed to be in 1937).
Of course all of this is highly debatable. What constitutes a "craze" is in need of a precise definition. However, from the elements above, I'd say that there is a hole between the time people ceased to believe in fairies, and the time they began to earnestly believe in aliens. This of course opens the question of what were they believing in, between the two World Wars ? Spiritualism is a candidate -- belief in communications with the dead, but dead humans, not deceased elves or martians. As I read in the relevant Wikipedia page:
In February 1921 Thomas Lynn Bradford committed suicide in an experiment designed to ascertain the existence of an afterlife. No further communications were received from him after this date.
That is what I call scientific dedication. And a clue that in 1921 some people were still quite crazy about spiritualism.